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Abstract

We survey results on the description of stochastically evolving genealogies of populationsand marked genealogies of multitype populations or spatial populations via tree-valued Markovprocesses on (marked) ultrametric measure spaces. In particular we explain the choice of statespaces and their topologies, describe the dynamics of genealogical Fleming-Viot and branchingmodels by well-posed martingale problems, and formulate the typical results on the longtimebehaviour. Furthermore we discuss the basic techniques of proofs and sketch as two key toolsof analysis the different forms of duality and the Girsanov transformation.